Riemannian manifold#Riemannian metrics and Riemannian manifolds

File:Georg_Friedrich_Bernhard_Riemann.jpeg.]]

In 1827, Carl Friedrich Gauss discovered that the Gaussian curvature of a surface embedded in 3-dimensional space only depends on local measurements made within the surface (the first fundamental form).{{sfn|do Carmo|1992|pp=35–36}} This result is known as the Theorema Egregium ("remarkable theorem" in Latin).

A map that preserves the local measurements of a surface is called a local isometry. A property of a surface is called an intrinsic property if it is preserved by local isometries and it is called an extrinsic property if it is not. In this language, the Theorema Egregium says that the Gaussian curvature is an intrinsic property of surfaces.

Riemannian manifolds and their curvature were first introduced non-rigorously by Bernhard Riemann in 1854.{{sfn|do Carmo|1992|p=37}} However, they would not be formalized until much later. In fact, the more primitive concept of a smooth manifold was first explicitly defined only in 1913 in a book by Hermann Weyl.{{sfn|do Carmo|1992|p=37}}

Élie Cartan introduced the Cartan connection, one of the first concepts of a connection. Levi-Civita defined the Levi-Civita connection, a special connection on a Riemannian manifold.

Albert Einstein used the theory of pseudo-Riemannian manifolds (a generalization of Riemannian manifolds) to develop general relativity. Specifically, the Einstein field equations are constraints on the curvature of spacetime, which is a 4-dimensional pseudo-Riemannian manifold.

File:Tangent plane to sphere with vectors.svg

Let $M$ be a smooth manifold. For each point $p\; \backslash in\; M$, there is an associated vector space $T\_pM$ called the tangent space of $M$ at $p$. Vectors in $T\_pM$ are thought of as the vectors tangent to $M$ at $p$.

However, $T\_pM$ does not come equipped with an inner product, a measuring stick that gives tangent vectors a concept of length and angle. This is an important deficiency because calculus teaches that to calculate the length of a curve, the length of vectors tangent to the curve must be defined. A Riemannian metric puts a measuring stick on every tangent space.

A Riemannian metric $g$ on $M$ assigns to each $p$ a positive-definite inner product $g\_p\; :\; T\_pM\; \backslash times\; T\_pM\; \backslash to\; \backslash mathbb\; R$ in a smooth way (see the section on regularity below).{{sfn|do Carmo|1992|p=38}} This induces a norm $\backslash |\backslash cdot\backslash |\_p\; :\; T\_pM\; \backslash to\; \backslash mathbb\; R$ defined by $\backslash |v\backslash |\_p\; =\; \backslash sqrt\{g\_p(v,v)\}$. A smooth manifold $M$ endowed with a Riemannian metric $g$ is a Riemannian manifold, denoted $(M,g)$.{{sfn|do Carmo|1992|p=38}} A Riemannian metric is a special case of a metric tensor.

A Riemannian metric is not to be confused with the distance function of a metric space, which is also called a metric.

If $(x^1,\backslash ldots,x^n):U\backslash to\backslash mathbb\{R\}^n$ are smooth local coordinates on $M$, the vectors

: $\backslash left\backslash \{\backslash frac\{\backslash partial\}\{\backslash partial\; x^1\}\backslash Big|\_p,\backslash dotsc,\; \backslash frac\{\backslash partial\}\{\backslash partial\; x^n\}\backslash Big|\_p\backslash right\backslash \}$

form a basis of the vector space $T\_pM$ for any $p\backslash in\; U$. Relative to this basis, one can define the Riemannian metric's components at each point $p$ by

: $g\_\{ij\}|\_p:=g\_p\backslash left(\backslash left.\backslash frac\{\backslash partial\; \}\{\backslash partial\; x^i\}\backslash right|\_p,\backslash left.\backslash frac\{\backslash partial\; \}\{\backslash partial\; x^j\}\backslash right|\_p\backslash right)$.{{sfn|Lee|2018|p=13}}

These $n^2$ functions $g\_\{ij\}:U\backslash to\backslash mathbb\{R\}$ can be put together into an $n\backslash times\; n$ matrix-valued function on $U$. The requirement that $g\_p$ is a positive-definite inner product then says exactly that this matrix-valued function is a symmetric positive-definite matrix at $p$.

In terms of the tensor algebra, the Riemannian metric can be written in terms of the dual basis $\backslash \{\; dx^1,\; \backslash ldots,\; dx^n\; \backslash \}$ of the cotangent bundle as

: $g=\backslash sum\_\{i,j\}g\_\{ij\}\; \backslash ,\; dx^i\; \backslash otimes\; dx^j.${{sfn|Lee|2018|p=13}}

The Riemannian metric $g$ is continuous if its components $g\_\{ij\}:U\backslash to\backslash mathbb\{R\}$ are continuous in any smooth coordinate chart $(U,x).$ The Riemannian metric $g$ is smooth if its components $g\_\{ij\}$ are smooth in any smooth coordinate chart. One can consider many other types of Riemannian metrics in this spirit, such as Lipschitz Riemannian metrics or measurable Riemannian metrics.

There are situations in geometric analysis in which one wants to consider non-smooth Riemannian metrics. See for instance (Gromov 1999) and (Shi and Tam 2002). However, in this article, $g$ is assumed to be smooth unless stated otherwise.

{{Main|Musical isomorphism}}

In analogy to how an inner product on a vector space induces an isomorphism between a vector space and its dual given by $v\; \backslash mapsto\; \backslash langle\; v,\; \backslash cdot\; \backslash rangle$, a Riemannian metric induces an isomorphism of bundles between the tangent bundle and the cotangent bundle. Namely, if $g$ is a Riemannian metric, then

: $(p,v)\; \backslash mapsto\; g\_p(v,\backslash cdot)$

is a isomorphism of smooth vector bundles from the tangent bundle $TM$ to the cotangent bundle $T^*M$.{{sfn|Lee|2018|p=26}}

An isometry is a function between Riemannian manifolds which preserves all of the structure of Riemannian manifolds. If two Riemannian manifolds have an isometry between them, they are called isometric, and they are considered to be the same manifold for the purpose of Riemannian geometry.

Specifically, if $(M,g)$ and $(N,h)$ are two Riemannian manifolds, a diffeomorphism $f:M\backslash to\; N$ is called an isometry if $g=f^\backslash ast\; h$,{{sfn|Lee|2018|p=12}} that is, if

: $g\_p(u,v)=h\_\{f(p)\}(df\_p(u),df\_p(v))$

for all $p\backslash in\; M$ and $u,v\backslash in\; T\_pM.$ For example, translations and rotations are both isometries from Euclidean space (to be defined soon) to itself.

One says that a smooth map $f:M\backslash to\; N,$ not assumed to be a diffeomorphism, is a local isometry if every $p\backslash in\; M$ has an open neighborhood $U$ such that $f:U\backslash to\; f(U)$ is an isometry (and thus a diffeomorphism).{{sfn|Lee|2018|p=12}}

An oriented $n$-dimensional Riemannian manifold $(M,g)$ has a unique $n$-form $dV\_g$ called the Riemannian volume form.{{sfn|Lee|2018|p=30}} The Riemannian volume form is preserved by orientation-preserving isometries.{{sfn|Lee|2018|p=31}} The volume form gives rise to a measure on $M$ which allows measurable functions to be integrated.{{citation needed|date=July 2024}} If $M$ is compact, the volume of $M$ is $\backslash int\_M\; dV\_g$.{{sfn|Lee|2018|p=30}}

Let $x^1,\backslash ldots,x^n$ denote the standard coordinates on $\backslash mathbb\{R\}^n.$ The (canonical) Euclidean metric $g^\backslash text\{can\}$ is given by{{sfn|Lee|2018|pp=12–13}}

: $g^\backslash text\{can\}\backslash left(\backslash sum\_i\; a\_i\; \backslash frac\{\backslash partial\}\{\backslash partial\; x^i\},\; \backslash sum\_j\; b\_j\; \backslash frac\{\backslash partial\}\{\backslash partial\; x^j\}\; \backslash right)\; =\; \backslash sum\_i\; a\_i\; b\_i$

or equivalently

: $g^\backslash text\{can\}\; =\; (dx^1)^2\; +\; \backslash cdots\; +\; (dx^n)^2$

or equivalently by its coordinate functions

: $g\_\{ij\}^\backslash text\{can\}\; =\; \backslash delta\_\{ij\}$ where $\backslash delta\_\{ij\}$ is the Kronecker delta

which together form the matrix

: $(g\_\{ij\}^\backslash text\{can\})\; =\; \backslash begin\{pmatrix\}$

1 & 0 & \cdots & 0 \\

0 & 1 & \cdots & 0 \\

\vdots & \vdots & \ddots & \vdots \\

0 & 0 & \cdots & 1

\end{pmatrix}.

The Riemannian manifold $(\backslash mathbb\{R\}^n,g^\backslash text\{can\})$ is called Euclidean space.

{{Main|Riemannian submanifold}}

File:Sphere filled blue.svg $S^n$ with the round metric is an embedded Riemannian submanifold of $\backslash mathbb\; R^\{n+1\}$.]]

Let $(M,g)$ be a Riemannian manifold and let $i\; :\; N\; \backslash to\; M$ be an immersed submanifold or an embedded submanifold of $M$. The pullback $i^*g$ of $g$ is a Riemannian metric on $N$, and $(N,\; i^*g)$ is said to be a Riemannian submanifold of $(M,g)$.{{sfn|Lee|2018|p=15}}

In the case where $N\; \backslash subseteq\; M$, the map $i\; :\; N\; \backslash to\; M$ is given by $i(x)\; =\; x$ and the metric $i^*g$ is just the restriction of $g$ to vectors tangent along $N$. In general, the formula for $i^*g$ is

: $i^*g\_p(v,w)\; =\; g\_\{i(p)\}\; \backslash big(\; di\_p(v),\; di\_p(w)\; \backslash big),$

where $di\_p(v)$ is the pushforward of $v$ by $i.$

Examples:

• The $n$-sphere
• : $S^n=\backslash \{x\backslash in\backslash mathbb\{R\}^\{n+1\}:(x^1)^2+\backslash cdots+(x^\{n+1\})^2=1\backslash \}$

:is a smooth embedded submanifold of Euclidean space $\backslash mathbb\; R^\{n+1\}$.{{sfn|Lee|2018|p=16}} The Riemannian metric this induces on $S^n$ is called the round metric or standard metric.

• Fix real numbers $a,b,c$. The ellipsoid
• :$\backslash left\backslash \{(x,y,z)\; \backslash in\; \backslash mathbb\; R^3\; :\; \backslash frac\{x^2\}\{a^2\}\; +\; \backslash frac\{y^2\}\{b^2\}\; +\; \backslash frac\{z^2\}\{c^2\}\; =\; 1\; \backslash right\backslash \}$

:is a smooth embedded submanifold of Euclidean space $\backslash mathbb\; R^3$.

• The graph of a smooth function $f:\backslash mathbb\{R\}^n\backslash to\backslash mathbb\{R\}$ is a smooth embedded submanifold of $\backslash mathbb\{R\}^\{n+1\}$ with its standard metric.
• If $(M,g)$ is not simply connected, there is a covering map $\backslash widetilde\{M\}\backslash to\; M$, where $\backslash widetilde\; M$ is the universal cover of $M$. This is an immersion (since it is locally a diffeomorphism), so $\backslash widetilde\; M$ automatically inherits a Riemannian metric. By the same principle, any smooth covering space of a Riemannian manifold inherits a Riemannian metric.

On the other hand, if $N$ already has a Riemannian metric $\backslash tilde\; g$, then the immersion (or embedding) $i\; :\; N\; \backslash to\; M$ is called an isometric immersion (or isometric embedding) if $\backslash tilde\; g\; =\; i^*\; g$. Hence isometric immersions and isometric embeddings are Riemannian submanifolds.{{sfn|Lee|2018|p=15}}

{{multiple image | image1=Grid for torus.svg | image2=Flat torus stereographic.svg | alt1=A 2x2 square grid | alt2=A torus embedded in Euclidean space | footer=A torus naturally carries a Euclidean metric, obtained by identifying opposite sides of a square (left). The resulting Riemannian manifold, called a flat torus, cannot be isometrically embedded in 3-dimensional Euclidean space (right), because it is necessary to bend and stretch the sheet in doing so. Thus the intrinsic geometry of a flat torus is different from that of an embedded torus.}}

Let $(M,g)$ and $(N,h)$ be two Riemannian manifolds, and consider the product manifold $M\backslash times\; N$. The Riemannian metrics $g$ and $h$ naturally put a Riemannian metric $\backslash widetilde\{g\}$ on $M\backslash times\; N,$ which can be described in a few ways.

• Considering the decomposition $T\_\{(p,q)\}(M\backslash times\; N)\; \backslash cong\; T\_pM\; \backslash oplus\; T\_qN,$ one may define
• : $\backslash widetilde\{g\}\_\{p,q\}\; ((u\_1,\; u\_2),\; (v\_1,\; v\_2))\; =\; g\_p(u\_1,\; v\_1)\; +\; h\_q(u\_2,\; v\_2).${{sfn|Lee|2018|p=20}}
• If $(U,x)$ is a smooth coordinate chart on $M$ and $(V,y)$ is a smooth coordinate chart on $N$, then $(U\; \backslash times\; V,\; (x,y))$ is a smooth coordinate chart on $M\; \backslash times\; N.$ Let $g\_U$ be the representation of $g$ in the chart $(U,x)$ and let $h\_V$ be the representation of $h$ in the chart $(V,y)$. The representation of $\backslash widetilde\{g\}$ in the coordinates $(U\; \backslash times\; V,(x,y))$ is
• :$\backslash widetilde\{g\}\; =\; \backslash sum\_\{ij\}\; \backslash widetilde\{g\}\_\{ij\}\; \backslash ,\; dx^i\; \backslash ,\; dx^j$ where {{sfn|Lee|2018|p=20}}

For example, consider the $n$-torus $T^n\; =\; S^1\backslash times\backslash cdots\backslash times\; S^1$. If each copy of $S^1$ is given the round metric, the product Riemannian manifold $T^n$ is called the flat torus. As another example, the Riemannian product $\backslash mathbb\; R\; \backslash times\; \backslash cdots\; \backslash times\; \backslash mathbb\; R$, where each copy of $\backslash mathbb\; R$ has the Euclidean metric, is isometric to $\backslash mathbb\; R^n$ with the Euclidean metric.

Let $g\_1,\; \backslash ldots,\; g\_k$ be Riemannian metrics on $M.$ If $f\_1,\; \backslash ldots,\; f\_k$ are any positive smooth functions on $M$, then $f\_1\; g\_1\; +\; \backslash ldots\; +\; f\_k\; g\_k$ is another Riemannian metric on $M.$

Theorem: Every smooth manifold admits a (non-canonical) Riemannian metric.{{sfn|Lee|2018|p=11}}

This is a fundamental result. Although much of the basic theory of Riemannian metrics can be developed using only that a smooth manifold is a locally Euclidean topological space, for this result it is necessary to use that smooth manifolds are Hausdorff and paracompact. The reason is that the proof makes use of a partition of unity.

{{Collapse top|title=Proof that every smooth manifold admits a Riemannian metric}}

Let $M$ be a smooth manifold and $\backslash \{(U\_\backslash alpha,\backslash varphi\_\backslash alpha)\backslash \}\_\{\backslash alpha\; \backslash in\; A\}$ a locally finite atlas so that $U\_\backslash alpha\; \backslash subseteq\; M$ are open subsets and $\backslash varphi\_\backslash alpha\; \backslash colon\; U\_\backslash alpha\backslash to\; \backslash varphi\_\backslash alpha(U\_\backslash alpha)\backslash subseteq\backslash mathbf\{R\}^n$ are diffeomorphisms. Such an atlas exists because the manifold is paracompact.

Let $\backslash \{\backslash tau\_\backslash alpha\backslash \}\_\{\backslash alpha\; \backslash in\; A\}$ be a differentiable partition of unity subordinate to the given atlas, i.e. such that $\backslash operatorname\{supp\}(\; \backslash tau\_\backslash alpha)\; \backslash subseteq\; U\_\backslash alpha$ for all $\backslash alpha\; \backslash in\; A$.

Define a Riemannian metric $g$ on $M$ by

: $g\; =\; \backslash sum\_\{\backslash alpha\; \backslash in\; A\}\; \backslash tau\_\backslash alpha\; \backslash cdot\; \backslash tilde\{g\}\_\backslash alpha$

where

:$\backslash tilde\{g\}\_\backslash alpha\; =\; \backslash varphi\_\backslash alpha^*\; g^\{\backslash text\{can\}\}.$

Here $g^\backslash text\{can\}$ is the Euclidean metric on $\backslash mathbb\; R^n$ and $\backslash varphi\_\backslash alpha^*g^\{\backslash mathrm\{can\}\}$ is its pullback along $\backslash varphi\_\backslash alpha$. While $\backslash tilde\{g\}\_\backslash alpha$ is only defined on $U\_\backslash alpha$, the product $\backslash tau\_\backslash alpha\; \backslash cdot\; \backslash tilde\{g\}\_\backslash alpha$ is defined and smooth on $M$ since $\backslash operatorname\{supp\}(\; \backslash tau\_\backslash alpha)\; \backslash subseteq\; U\_\backslash alpha$. It takes the value 0 outside of $U\_\backslash alpha$. Because the atlas is locally finite, at every point the sum contains only finitely many nonzero terms, so the sum converges. It is straightforward to check that $g$ is a Riemannian metric.

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An alternative proof uses the Whitney embedding theorem to embed $M$ into Euclidean space and then pulls back the metric from Euclidean space to $M$. On the other hand, the Nash embedding theorem states that, given any smooth Riemannian manifold $(M,g),$ there is an embedding $F:M\backslash to\backslash mathbb\{R\}^N$ for some $N$ such that the pullback by $F$ of the standard Riemannian metric on $\backslash mathbb\{R\}^N$ is $g.$ That is, the entire structure of a smooth Riemannian manifold can be encoded by a diffeomorphism to a certain embedded submanifold of some Euclidean space. Therefore, one could argue that nothing can be gained from the consideration of abstract smooth manifolds and their Riemannian metrics. However, there are many natural smooth Riemannian manifolds, such as the set of rotations of three-dimensional space and hyperbolic space, of which any representation as a submanifold of Euclidean space will fail to represent their remarkable symmetries and properties as clearly as their abstract presentations do.

An admissible curve is a piecewise smooth curve $\backslash gamma\; :\; [0,1]\; \backslash to\; M$ whose velocity $\backslash gamma\text{'}(t)\; \backslash in\; T\_\{\backslash gamma(t)\}M$ is nonzero everywhere it is defined. The nonnegative function $t\backslash mapsto\backslash |\backslash gamma\text{'}(t)\backslash |\_\{\backslash gamma(t)\}$ is defined on the interval $[0,1]$ except for at finitely many points. The length $L(\backslash gamma)$ of an admissible curve $\backslash gamma\; :\; [0,1]\; \backslash to\; M$ is defined as

: $L(\backslash gamma)=\backslash int\_0^1\; \backslash |\backslash gamma\text{'}(t)\backslash |\_\{\backslash gamma(t)\}\; \backslash ,\; dt.$

The integrand is bounded and continuous except at finitely many points, so it is integrable. For $(M,g)$ a connected Riemannian manifold, define $d\_g:M\backslash times\; M\backslash to[0,\backslash infty)$ by

: $d\_g(p,q)\; =\; \backslash inf\; \backslash \{\; L(\backslash gamma)\; :\; \backslash gamma\; \backslash text\{\; an\; admissible\; curve\; with\; \}\; \backslash gamma(0)\; =\; p,\; \backslash gamma(1)\; =\; q\; \backslash \}.$

Theorem: $(M,d\_g)$ is a metric space, and the metric topology on $(M,d\_g)$ coincides with the topology on $M$.{{sfn|Lee|2018|p=39}}

{{Collapse top|title=Proof sketch that $(M,d\_g)$ is a metric space, and the metric topology on $(M,d\_g)$ agrees with the topology on $M$}}

In verifying that $(M,d\_g)$ satisfies all of the axioms of a metric space, the most difficult part is checking that $p\backslash neq\; q$ implies $d\_g(p,q)>0$. Verification of the other metric space axioms is omitted.

There must be some precompact open set around p which every curve from p to q must escape. By selecting this open set to be contained in a coordinate chart, one can reduce the claim to the well-known fact that, in Euclidean geometry, the shortest curve between two points is a line. In particular, as seen by the Euclidean geometry of a coordinate chart around p, any curve from p to q must first pass though a certain "inner radius." The assumed continuity of the Riemannian metric g only allows this "coordinate chart geometry" to distort the "true geometry" by some bounded factor.

To be precise, let $(U,x)$ be a smooth coordinate chart with $x(p)=0$ and $q\backslash notin\; U.$ Let $V\backslash ni\; x$ be an open subset of $U$ with $\backslash overline\{V\}\backslash subset\; U.$ By continuity of $g$ and compactness of $\backslash overline\{V\},$ there is a positive number $\backslash lambda$ such that $g(X,X)\backslash geq\backslash lambda\backslash |X\backslash |^2$ for any $r\backslash in\; V$ and any $X\backslash in\; T\_rM,$ where $\backslash |\backslash cdot\backslash |$ denotes the Euclidean norm induced by the local coordinates. Let R denote $\backslash sup\backslash \{r>0:B\_r(0)\backslash subset\; x(V)\backslash \}$.

Now, given any admissible curve $\backslash gamma:[0,1]\backslash to\; M$ from p to q, there must be some minimal $\backslash delta>0$ such that $\backslash gamma(\backslash delta)\backslash notin\; V;$ clearly $\backslash gamma(\backslash delta)\backslash in\backslash partial\; V.$

The length of $\backslash gamma$ is at least as large as the restriction of $\backslash gamma$ to $[0,\backslash delta].$ So

: $L(\backslash gamma)\backslash geq\backslash sqrt\{\backslash lambda\}\backslash int\_0^\backslash delta\backslash |\backslash gamma\text{'}(t)\backslash |\backslash ,dt.$

The integral which appears here represents the Euclidean length of a curve from 0 to $x(\backslash partial\; V)\backslash subset\backslash mathbb\{R\}^n$, and so it is greater than or equal to R. So we conclude $L(\backslash gamma)\backslash geq\backslash sqrt\{\backslash lambda\}R.$

The observation about comparison between lengths measured by g and Euclidean lengths measured in a smooth coordinate chart, also verifies that the metric space topology of $(M,d\_g)$ coincides with the original topological space structure of $M$.

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Although the length of a curve is given by an explicit formula, it is generally impossible to write out the distance function $d\_g$ by any explicit means. In fact, if $M$ is compact, there always exist points where $d\_g:M\backslash times\; M\backslash to\backslash mathbb\{R\}$ is non-differentiable, and it can be remarkably difficult to even determine the location or nature of these points, even in seemingly simple cases such as when $(M,g)$ is an ellipsoid.{{citation needed|date=July 2024}}

If one works with Riemannian metrics that are merely continuous but possibly not smooth, the length of an admissible curve and the Riemannian distance function are defined exactly the same, and, as before, $(M,d\_g)$ is a metric space and the metric topology on $(M,d\_g)$ coincides with the topology on $M$.{{sfn|Burtscher|2015|p=276}}

The diameter of the metric space $(M,d\_g)$ is

: $\backslash operatorname\{diam\}(M,d\_g)=\backslash sup\backslash \{d\_g(p,q):p,q\backslash in\; M\backslash \}.$

The Hopf–Rinow theorem shows that if $(M,d\_g)$ is complete and has finite diameter, it is compact. Conversely, if $(M,d\_g)$ is compact, then the function $d\_g:M\backslash times\; M\backslash to\backslash mathbb\{R\}$ has a maximum, since it is a continuous function on a compact metric space. This proves the following.

: If $(M,d\_g)$ is complete, then it is compact if and only if it has finite diameter.

This is not the case without the completeness assumption; for counterexamples one could consider any open bounded subset of a Euclidean space with the standard Riemannian metric. It is also not true that any complete metric space of finite diameter must be compact; it matters that the metric space came from a Riemannian manifold.

{{Main|Affine connection}}

An (affine) connection is an additional structure on a Riemannian manifold that defines differentiation of one vector field with respect to another. Connections contain geometric data, and two Riemannian manifolds with different connections have different geometry.

Let $\backslash mathfrak\; X(M)$ denote the space of vector fields on $M$. An (affine) connection

: $\backslash nabla\; :\; \backslash mathfrak\; X(M)\; \backslash times\; \backslash mathfrak\; X(M)\; \backslash to\; \backslash mathfrak\; X(M)$

on $M$ is a bilinear map

$(X,Y)\; \backslash mapsto\; \backslash nabla\_X\; Y$ such that

1. For every function $f\; \backslash in\; C^\backslash infty(M)$, $\backslash nabla\_\{f\_1\; X\_1\; +\; f\_2\; X\_2\}\; Y\; =\; f\_1\; \backslash ,\backslash nabla\_\{X\_1\}\; Y\; +\; f\_2\; \backslash ,\; \backslash nabla\_\{X\_2\}\; Y,$
2. The product rule $\backslash nabla\_X\; fY=X(f)Y+\; f\backslash ,\backslash nabla\_X\; Y$ holds.{{sfn|Lee|2018|pp=89–91}}

The expression $\backslash nabla\_X\; Y$ is called the covariant derivative of $Y$ with respect to $X$.

{{Main|Levi-Civita connection}}

Two Riemannian manifolds with different connections have different geometry. Thankfully, there is a natural connection associated to a Riemannian manifold called the Levi-Civita connection.

A connection $\backslash nabla$ is said to preserve the metric if

: $X\backslash bigl(g(Y,Z)\backslash bigr)\; =\; g(\backslash nabla\_X\; Y,\; Z)\; +\; g(Y,\; \backslash nabla\_X\; Z)$

A connection $\backslash nabla$ is torsion-free if

: $\backslash nabla\_X\; Y\; -\; \backslash nabla\_Y\; X\; =\; [X,Y],$

where $[\backslash cdot,\backslash cdot]$ is the Lie bracket.

A Levi-Civita connection is a torsion-free connection that preserves the metric. Once a Riemannian metric is fixed, there exists a unique Levi-Civita connection.{{sfn|Lee|2018|pp=122–123}} Note that the definition of preserving the metric uses the regularity of $g$.

If $\backslash gamma\; :\; [0,1]\; \backslash to\; M$ is a smooth curve, a smooth vector field along $\backslash gamma$ is a smooth map $X\; :\; [0,1]\; \backslash to\; TM$ such that $X(t)\; \backslash in\; T\_\{\backslash gamma(t)\}M$ for all $t\; \backslash in\; [0,1]$. The set $\backslash mathfrak\; X(\backslash gamma)$ of smooth vector fields along $\backslash gamma$ is a vector space under pointwise vector addition and scalar multiplication.{{sfn|Lee|2018|p=100}} One can also pointwise multiply a smooth vector field along $\backslash gamma$ by a smooth function $f\; :\; [0,1]\; \backslash to\; \backslash mathbb\; R$:

: $(fX)(t)\; =\; f(t)X(t)$ for $X\; \backslash in\; \backslash mathfrak\; X(\backslash gamma).$

Let $X$ be a smooth vector field along $\backslash gamma$. If $\backslash tilde\; X$ is a smooth vector field on a neighborhood of the image of $\backslash gamma$ such that $X(t)\; =\; \backslash tilde\; X\_\{\backslash gamma(t)\}$, then $\backslash tilde\; X$ is called an extension of $X$.

Given a fixed connection $\backslash nabla$ on $M$ and a smooth curve $\backslash gamma\; :\; [0,1]\; \backslash to\; M$, there is a unique operator $D\_t\; :\; \backslash mathfrak\; X(\backslash gamma)\; \backslash to\; \backslash mathfrak\; X(\backslash gamma)$, called the covariant derivative along $\backslash gamma$, such that:{{sfn|Lee|2018|pp=101–102}}

1. $D\_t(aX+bY)\; =\; a\backslash ,D\_tX\; +\; b\backslash ,D\_tY,$
2. $D\_t(fX)\; =\; f\text{'}X\; +\; f\backslash ,D\_tX,$
3. If $\backslash tilde\; X$ is an extension of $X$, then $D\_tX(t)\; =\; \backslash nabla\_\{\backslash gamma\text{'}(t)\}\; \backslash tilde\; X$.

{{Main|Geodesic}}

{{multiple image | image1=Plane geodesic.svg | image2=Sphere geodesic.svg | footer=In Euclidean space $\backslash mathbb\; R^n$ (left), the maximal geodesics are straight lines. In the round sphere $S^n$ (right), the maximal geodesics are great circles.}}

Geodesics are curves with no intrinsic acceleration. Equivalently, geodesics are curves that locally take the shortest path between two points. They are the generalization of straight lines in Euclidean space to arbitrary Riemannian manifolds. An ant living in a Riemannian manifold walking straight ahead without making any effort to accelerate or turn would trace out a geodesic.

Fix a connection $\backslash nabla$ on $M$. Let $\backslash gamma\; :\; [0,1]\; \backslash to\; M$ be a smooth curve. The acceleration of $\backslash gamma$ is the vector field $D\_t\backslash gamma\text{'}$ along $\backslash gamma$. If $D\_t\backslash gamma\text{'}\; =\; 0$ for all $t$, $\backslash gamma$ is called a geodesic.{{sfn|Lee|2018|p=103}}

For every $p\; \backslash in\; M$ and $v\; \backslash in\; T\_pM$, there exists a geodesic $\backslash gamma\; :\; I\; \backslash to\; M$ defined on some open interval $I$ containing 0 such that $\backslash gamma(0)\; =\; p$ and $\backslash gamma\text{'}(0)\; =\; v$. Any two such geodesics agree on their common domain.{{sfn|Lee|2018|pp=103–104}} Taking the union over all open intervals $I$ containing 0 on which a geodesic satisfying $\backslash gamma(0)\; =\; p$ and $\backslash gamma\text{'}(0)\; =\; v$ exists, one obtains a geodesic called a maximal geodesic of which every geodesic satisfying $\backslash gamma(0)\; =\; p$ and $\backslash gamma\text{'}(0)\; =\; v$ is a restriction.{{sfn|Lee|2018|p=105}}

Every curve $\backslash gamma\; :\; [0,1]\; \backslash to\; M$ that has the shortest length of any admissible curve with the same endpoints as $\backslash gamma$ is a geodesic (in a unit-speed reparameterization).{{sfn|Lee|2018|p=156}}

• The nonconstant maximal geodesics of the Euclidean plane $\backslash mathbb\; R^2$ are exactly the straight lines.{{sfn|Lee|2018|p=105}} This agrees with the fact from Euclidean geometry that the shortest path between two points is a straight line segment.
• The nonconstant maximal geodesics of $S^2$ with the round metric are exactly the great circles.{{sfn|Lee|2018|p=137}} Since the Earth is approximately a sphere, this means that the shortest path a plane can fly between two locations on Earth is a segment of a great circle.

{{Main|Hopf–Rinow theorem}}

File:Punctured plane is not geodesically complete.svg

The Riemannian manifold $M$ with its Levi-Civita connection is geodesically complete if the domain of every maximal geodesic is $(-\backslash infty,\backslash infty)$.{{sfn|Lee|2018|p=131}} The plane $\backslash mathbb\; R^2$ is geodesically complete. On the other hand, the punctured plane $\backslash mathbb\{R\}^2\backslash smallsetminus\backslash \{(0,0)\backslash \}$ with the restriction of the Riemannian metric from $\backslash mathbb\; R^2$ is not geodesically complete as the maximal geodesic with initial conditions $p\; =\; (1,1)$, $v\; =\; (1,1)$ does not have domain $\backslash mathbb\; R$.

The Hopf–Rinow theorem characterizes geodesically complete manifolds.

Theorem: Let $(M,g)$ be a connected Riemannian manifold. The following are equivalent:{{sfn|do Carmo|1992|pp=146–147}}

• The metric space $(M,d\_g)$ is complete (every $d\_g$-Cauchy sequence converges),
• All closed and bounded subsets of $M$ are compact,
• $M$ is geodesically complete.

{{Main|Parallel transport}}

File:Parallel transport sphere2.svg

In Euclidean space, all tangent spaces are canonically identified with each other via translation, so it is easy to move vectors from one tangent space to another. Parallel transport is a way of moving vectors from one tangent space to another along a curve in the setting of a general Riemannian manifold. Given a fixed connection, there is a unique way to do parallel transport.{{sfn|Lee|2018|pp=105–110}}

Specifically, call a smooth vector field $V$ along a smooth curve $\backslash gamma$ parallel along $\backslash gamma$ if $D\_t\; V\; =\; 0$ identically.{{sfn|Lee|2018|p=105}} Fix a curve $\backslash gamma\; :\; [0,1]\; \backslash to\; M$ with $\backslash gamma(0)\; =\; p$ and $\backslash gamma(1)\; =\; q$. to parallel transport a vector $v\; \backslash in\; T\_pM$ to a vector in $T\_qM$ along $\backslash gamma$, first extend $v$ to a vector field parallel along $\backslash gamma$, and then take the value of this vector field at $q$.

The images below show parallel transport induced by the Levi-Civita connection associated to two different Riemannian metrics on the punctured plane $\backslash mathbb\; R^2\; \backslash smallsetminus\; \backslash \{0,0\backslash \}$. The curve the parallel transport is done along is the unit circle. In polar coordinates, the metric on the left is the standard Euclidean metric $dx^2\; +\; dy^2\; =\; dr^2\; +\; r^2\; \backslash ,\; d\backslash theta^2$, while the metric on the right is $dr^2\; +\; d\backslash theta^2$. This second metric has a singularity at the origin, so it does not extend past the puncture, but the first metric extends to the entire plane.

{{multiple image

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| header = Parallel transports on the punctured plane under Levi-Civita connections

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|image1=Cartesian_transport.gif

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|caption1=This transport is given by the metric $dr^2\; +\; r^2\; d\backslash theta^2$.

|alt1=Cartesian transport

|image2=Circle_transport.gif

|width2=200

|caption2=This transport is given by the metric $dr^2\; +\; d\backslash theta^2$.

|alt2=Polar transport

}}

Warning: This is parallel transport on the punctured plane along the unit circle, not parallel transport on the unit circle. Indeed, in the first image, the vectors fall outside of the tangent space to the unit circle.

{{Main|Riemann curvature tensor}}

The Riemann curvature tensor measures precisely the extent to which parallel transporting vectors around a small rectangle is not the identity map.{{sfn|Lee|2018|p=201}} The Riemann curvature tensor is 0 at every point if and only if the manifold is locally isometric to Euclidean space.{{sfn|Lee|2018|p=200}}

Fix a connection $\backslash nabla$ on $M$. The Riemann curvature tensor is the map $R\; :\; \backslash mathfrak\; X(M)\; \backslash times\; \backslash mathfrak\; X(M)\; \backslash times\; \backslash mathfrak\; X(M)\; \backslash to\; \backslash mathfrak\; X(M)$ defined by

:$R(X,\; Y)Z\; =\; \backslash nabla\_X\backslash nabla\_Y\; Z\; -\; \backslash nabla\_Y\; \backslash nabla\_X\; Z\; -\; \backslash nabla\_\{[X,\; Y]\}\; Z$

where $[X,\; Y]$ is the Lie bracket of vector fields. The Riemann curvature tensor is a $(1,3)$-tensor field.{{sfn|Lee|2018|pp=196–197}}

{{Main|Ricci curvature}}

Fix a connection $\backslash nabla$ on $M$. The Ricci curvature tensor is

: $Ric(X,Y)\; =\; \backslash operatorname\{tr\}(Z\; \backslash mapsto\; R(Z,X)Y)$

where $\backslash operatorname\{tr\}$ is the trace. The Ricci curvature tensor is a covariant 2-tensor field.{{sfn|Lee|2018|p=207}}

{{Main|Einstein manifold}}

The Ricci curvature tensor $Ric$ plays a defining role in the theory of Einstein manifolds, which has applications to the study of gravity. A (pseudo-)Riemannian metric $g$ is called an Einstein metric if Einstein's equation

: $Ric\; =\; \backslash lambda\; g$ for some constant $\backslash lambda$

holds, and a (pseudo-)Riemannian manifold whose metric is Einstein is called an Einstein manifold.{{sfn|Lee|2018|p=210}} Examples of Einstein manifolds include Euclidean space, the $n$-sphere, hyperbolic space, and complex projective space with the Fubini-Study metric.

{{Main|Scalar curvature}}

A Riemannian manifold is said to have constant curvature {{mvar|κ}} if every sectional curvature equals the number {{mvar|κ}}. This is equivalent to the condition that, relative to any coordinate chart, the Riemann curvature tensor can be expressed in terms of the metric tensor as

:$R\_\{ijkl\}=\backslash kappa(g\_\{il\}g\_\{jk\}-g\_\{ik\}g\_\{jl\}).$

This implies that the Ricci curvature is given by {{math|R_jk {{=}} (n − 1)κg_jk}} and the scalar curvature is {{math|n(n − 1)κ}}, where {{mvar|n}} is the dimension of the manifold. In particular, every Riemannian manifold of constant curvature is an Einstein manifold, thereby having constant scalar curvature. As found by Bernhard Riemann in his 1854 lecture introducing Riemannian geometry, the locally-defined Riemannian metric

:$\backslash frac\{dx\_1^2+\backslash cdots+dx\_n^2\}\{(1+\backslash frac\{\backslash kappa\}\{4\}(x\_1^2+\backslash cdots+x\_n^2))^2\}$

has constant curvature {{mvar|κ}}. Any two Riemannian manifolds of the same constant curvature are locally isometric, and so it follows that any Riemannian manifold of constant curvature {{mvar|κ}} can be covered by coordinate charts relative to which the metric has the above form.{{sfnm|1a1=Wolf|1y=2011|1loc=Chapter 2}}

A Riemannian space form is a Riemannian manifold with constant curvature which is additionally connected and geodesically complete. A Riemannian space form is said to be a spherical space form if the curvature is positive, a Euclidean space form if the curvature is zero, and a hyperbolic space form or hyperbolic manifold if the curvature is negative. In any dimension, the sphere with its standard Riemannian metric, Euclidean space, and hyperbolic space are Riemannian space forms of constant curvature {{math|1}}, {{math|0}}, and {{math|−1}} respectively. Furthermore, the Killing–Hopf theorem says that any simply-connected spherical space form is homothetic to the sphere, any simply-connected Euclidean space form is homothetic to Euclidean space, and any simply-connected hyperbolic space form is homothetic to hyperbolic space.{{sfnm|1a1=Wolf|1y=2011|1loc=Chapter 2}}

Using the covering manifold construction, any Riemannian space form is isometric to the quotient manifold of a simply-connected Riemannian space form, modulo a certain group action of isometries. For example, the isometry group of the {{mvar|n}}-sphere is the orthogonal group {{math|O(n + 1)}}. Given any finite subgroup {{mvar|G}} thereof in which only the identity matrix possesses {{math|1}} as an eigenvalue, the natural group action of the orthogonal group on the {{mvar|n}}-sphere restricts to a group action of {{mvar|G}}, with the quotient manifold {{math|Sⁿ / G}} inheriting a geodesically complete Riemannian metric of constant curvature {{math|1}}. Up to homothety, every spherical space form arises in this way; this largely reduces the study of spherical space forms to problems in group theory. For instance, this can be used to show directly that every even-dimensional spherical space form is homothetic to the standard metric on either the sphere or real projective space. There are many more odd-dimensional spherical space forms, although there are known algorithms for their classification. The list of three-dimensional spherical space forms is infinite but explicitly known, and includes the lens spaces and the Poincaré dodecahedral space.{{sfnm|1a1=Wolf|1y=2011|1loc=Chapters 2 and 7}}

The case of Euclidean and hyperbolic space forms can likewise be reduced to group theory, based on study of the isometry group of Euclidean space and hyperbolic space. For example, the class of two-dimensional Euclidean space forms includes Riemannian metrics on the Klein bottle, the Möbius strip, the torus, the cylinder {{math|S¹ × ℝ}}, along with the Euclidean plane. Unlike the case of two-dimensional spherical space forms, in some cases two space form structures on the same manifold are not homothetic. The case of two-dimensional hyperbolic space forms is even more complicated, having to do with Teichmüller space. In three dimensions, the Euclidean space forms are known, while the geometry of hyperbolic space forms in three and higher dimensions remains an area of active research known as hyperbolic geometry.{{sfnm|1a1=Wolf|1y=2011|1loc=Chapters 2 and 3}}

Let {{mvar|G}} be a Lie group, such as the group of rotations in three-dimensional space. Using the group structure, any inner product on the tangent space at the identity (or any other particular tangent space) can be transported to all other tangent spaces to define a Riemannian metric. Formally, given an inner product {{math|g_e}} on the tangent space at the identity, the inner product on the tangent space at an arbitrary point {{mvar|p}} is defined by

:$g\_p(u,v)=g\_e(dL\_\{p^\{-1\}\}(u),dL\_\{p^\{-1\}\}(v)),$

where for arbitrary {{mvar|x}}, {{math|L_x}} is the left multiplication map {{math|G → G}} sending a point {{mvar|y}} to {{math|xy}}. Riemannian metrics constructed this way are left-invariant; right-invariant Riemannian metrics could be constructed likewise using the right multiplication map instead.

The Levi-Civita connection and curvature of a general left-invariant Riemannian metric can be computed explicitly in terms of {{math|g_e}}, the adjoint representation of {{mvar|G}}, and the Lie algebra associated to {{mvar|G}}.{{sfnm|1a1=Cheeger|1a2=Ebin|1y=2008|1loc=Proposition 3.18}} These formulas simplify considerably in the special case of a Riemannian metric which is bi-invariant (that is, simultaneously left- and right-invariant).{{sfnm|1a1=Cheeger|1a2=Ebin|1y=2008|1loc=Corollary 3.19|2a1=Petersen|2y=2016|2loc=Section 4.4}} All left-invariant metrics have constant scalar curvature.

Left- and bi-invariant metrics on Lie groups are an important source of examples of Riemannian manifolds. Berger spheres, constructed as left-invariant metrics on the special unitary group SU(2), are among the simplest examples of the collapsing phenomena, in which a simply-connected Riemannian manifold can have small volume without having large curvature.{{sfnm|1a1=Petersen|1y=2016|1loc=Section 4.4.3 and p. 399}} They also give an example of a Riemannian metric which has constant scalar curvature but which is not Einstein, or even of parallel Ricci curvature.{{sfnm|1a1=Petersen|1y=2016|1p=369}} Hyperbolic space can be given a Lie group structure relative to which the metric is left-invariant.In the upper half-space model of hyperbolic space, the Lie group structure is defined by $(x\_1,\backslash ldots,x\_n)\backslash cdot(y\_1,\backslash ldots,y\_n)=(x\_1+y\_nx\_1,\backslash ldots,x\_\{n-1\}+y\_nx\_\{n-1\},x\_ny\_n).${{sfnm|1a1=Lee|1y=2018|1loc=Example 3.16f}} Any bi-invariant Riemannian metric on a Lie group has nonnegative sectional curvature, giving a variety of such metrics: a Lie group can be given a bi-invariant Riemannian metric if and only if it is the product of a compact Lie group with an abelian Lie group.{{sfnm|1a1=Lee|1y=2018|1p=72|2a1=Milnor|2y=1976}}

A Riemannian manifold {{math|(M, g)}} is said to be homogeneous if for every pair of points {{mvar|x}} and {{mvar|y}} in {{mvar|M}}, there is some isometry {{mvar|f}} of the Riemannian manifold sending {{mvar|x}} to {{mvar|y}}. This can be rephrased in the language of group actions as the requirement that the natural action of the isometry group is transitive. Every homogeneous Riemannian manifold is geodesically complete and has constant scalar curvature.{{sfnm|1a1=Kobayashi|1a2=Nomizu|1y=1963|1loc=Theorem IV.4.5}}

Up to isometry, all homogeneous Riemannian manifolds arise by the following construction. Given a Lie group {{mvar|G}} with compact subgroup {{mvar|K}} which does not contain any nontrivial normal subgroup of {{mvar|G}}, fix any complemented subspace {{mvar|W}} of the Lie algebra of {{mvar|K}} within the Lie algebra of {{mvar|G}}. If this subspace is invariant under the linear map {{math|ad_G(k): W → W}} for any element {{mvar|k}} of {{mvar|K}}, then {{mvar|G}}-invariant Riemannian metrics on the coset space {{math|G/K}} are in one-to-one correspondence with those inner products on {{mvar|W}} which are invariant under {{math|ad_G(k): W → W}} for every element {{mvar|k}} of {{mvar|K}}.{{sfnm|1a1=Besse|1y=1987|1loc=Section 7C}} Each such Riemannian metric is homogeneous, with {{mvar|G}} naturally viewed as a subgroup of the full isometry group.

The above example of Lie groups with left-invariant Riemannian metrics arises as a very special case of this construction, namely when {{mvar|K}} is the trivial subgroup containing only the identity element. The calculations of the Levi-Civita connection and the curvature referenced there can be generalized to this context, where now the computations are formulated in terms of the inner product on {{mvar|W}}, the Lie algebra of {{mvar|G}}, and the direct sum decomposition of the Lie algebra of {{mvar|G}} into the Lie algebra of {{mvar|K}} and {{mvar|W}}.{{sfnm|1a1=Besse|1y=1987|1loc=Section 7C}} This reduces the study of the curvature of homogeneous Riemannian manifolds largely to algebraic problems. This reduction, together with the flexibility of the above construction, makes the class of homogeneous Riemannian manifolds very useful for constructing examples.

{{Main|Symmetric space}}

A connected Riemannian manifold {{math|(M, g)}} is said to be symmetric if for every point {{mvar|p}} of {{mvar|M}} there exists some isometry of the manifold with {{mvar|p}} as a fixed point and for which the negation of the differential at {{mvar|p}} is the identity map. Every Riemannian symmetric space is homogeneous, and consequently is geodesically complete and has constant scalar curvature. However, Riemannian symmetric spaces also have a much stronger curvature property not possessed by most homogeneous Riemannian manifolds, namely that the Riemann curvature tensor and Ricci curvature are parallel. Riemannian manifolds with this curvature property, which could loosely be phrased as "constant Riemann curvature tensor" (not to be confused with constant curvature), are said to be locally symmetric. This property nearly characterizes symmetric spaces; Élie Cartan proved in the 1920s that a locally symmetric Riemannian manifold which is geodesically complete and simply-connected must in fact be symmetric.{{sfnm|1a1=Petersen|1y=2016|1loc=Chapter 10}}

Many of the fundamental examples of Riemannian manifolds are symmetric. The most basic include the sphere and real projective spaces with their standard metrics, along with hyperbolic space. The complex projective space, quaternionic projective space, and Cayley plane are analogues of the real projective space which are also symmetric, as are complex hyperbolic space, quaternionic hyperbolic space, and Cayley hyperbolic space, which are instead analogues of hyperbolic space. Grassmannian manifolds also carry natural Riemannian metrics making them into symmetric spaces. Among the Lie groups with left-invariant Riemannian metrics, those which are bi-invariant are symmetric.{{sfnm|1a1=Petersen|1y=2016|1loc=Chapter 10}}

Based on their algebraic formulation as special kinds of homogeneous spaces, Cartan achieved an explicit classification of symmetric spaces which are irreducible, referring to those which cannot be locally decomposed as product spaces. Every such space is an example of an Einstein manifold; among them only the one-dimensional manifolds have zero scalar curvature. These spaces are important from the perspective of Riemannian holonomy. As found in the 1950s by Marcel Berger, any Riemannian manifold which is simply-connected and irreducible is either a symmetric space or has Riemannian holonomy belonging to a list of only seven possibilities. Six of the seven exceptions to symmetric spaces in Berger's classification fall into the fields of Kähler geometry, quaternion-Kähler geometry, G₂ geometry, and Spin(7) geometry, each of which study Riemannian manifolds equipped with certain extra structures and symmetries. The seventh exception is the study of 'generic' Riemannian manifolds with no particular symmetry, as reflected by the maximal possible holonomy group.{{sfnm|1a1=Petersen|1y=2016|1loc=Chapter 10}}

{{No footnotes|section|date=July 2024}}

The statements and theorems above are for finite-dimensional manifolds—manifolds whose charts map to open subsets of $\backslash R^n.$ These can be extended, to a certain degree, to infinite-dimensional manifolds; that is, manifolds that are modeled after a topological vector space; for example, Fréchet, Banach, and Hilbert manifolds.

Riemannian metrics are defined in a way similar to the finite-dimensional case. However, there is a distinction between two types of Riemannian metrics:

• A weak Riemannian metric on $M$ is a smooth function $g\; :\; TM\; \backslash times\; TM\; \backslash to\; \backslash R,$ such that for any $x\; \backslash in\; M$ the restriction $g\_x\; :\; T\_xM\; \backslash times\; T\_xM\; \backslash to\; \backslash R$ is an inner product on $T\_xM.${{citation needed|date=July 2024}}
• A strong Riemannian metric on $M$ is a weak Riemannian metric such that $g\_x$ induces the topology on $T\_xM$. If $g$ is a strong Riemannian metric, then $M$ must be a Hilbert manifold.{{citation needed|date=July 2024}}

• If $(H,\; \backslash langle\; \backslash ,\backslash cdot,\; \backslash cdot\backslash ,\; \backslash rangle)$ is a Hilbert space, then for any $x\; \backslash in\; H,$ one can identify $H$ with $T\_xH.$ The metric $g\_x(u,v)\; =\; \backslash langle\; u,\; v\; \backslash rangle$ for all $x,\; u,\; v\; \backslash in\; H$ is a strong Riemannian metric.{{citation needed|date=July 2024}}
• Let $(M,\; g)$ be a compact Riemannian manifold and denote by $\backslash operatorname\{Diff\}(M)$ its diffeomorphism group. The latter is a smooth manifold (see here) and in fact, a Lie group.{{citation needed|date=July 2024}} Its tangent bundle at the identity is the set of smooth vector fields on $M.${{citation needed|date=July 2024}} Let $\backslash mu$ be a volume form on $M.$ The $L^2$ weak Riemannian metric on $\backslash operatorname\{Diff\}(M)$, denoted $G$, is defined as follows. Let $f\backslash in\; \backslash operatorname\{Diff\}(M),$ $u,\; v\; \backslash in\; T\_f\backslash operatorname\{Diff\}(M).$ Then for $x\; \backslash in\; M,\; u(x)\; \backslash in\; T\_\{f(x)\}M$,
• :$G\_f(u,v)\; =\; \backslash int\; \_M\; g\_\{f(x)\}\; (u(x),v(x))\; \backslash ,\; d\backslash mu\; (x)$.{{citation needed|date=July 2024}}

Length of curves and the Riemannian distance function $d\_g\; :\; M\; \backslash times\; M\; \backslash to\; [0,\backslash infty)$ are defined in a way similar to the finite-dimensional case. The distance function $d\_g$, called the geodesic distance, is always a pseudometric (a metric that does not separate points), but it may not be a metric.{{sfn|Magnani|Tiberio|2020}} In the finite-dimensional case, the proof that the Riemannian distance function separates points uses the existence of a pre-compact open set around any point. In the infinite case, open sets are no longer pre-compact, so the proof fails.

• If $g$ is a strong Riemannian metric on $M$, then $d\_g$ separates points (hence is a metric) and induces the original topology.{{citation needed|date=July 2024}}
• If $g$ is a weak Riemannian metric, $d\_g$ may fail to separate points. In fact, it may even be identically 0.{{sfn|Magnani|Tiberio|2020}} For example, if $(M,\; g)$ is a compact Riemannian manifold, then the $L^2$ weak Riemannian metric on $\backslash operatorname\{Diff\}(M)$ induces vanishing geodesic distance.{{sfn|Michor|Mumford|2005}}

In the case of strong Riemannian metrics, one part of the finite-dimensional Hopf–Rinow still holds.

Theorem: Let $(M,\; g)$ be a strong Riemannian manifold. Then metric completeness (in the metric $d\_g$) implies geodesic completeness.{{citation needed|date=July 2024}}

However, a geodesically complete strong Riemannian manifold might not be metrically complete and it might have closed and bounded subsets that are not compact.{{citation needed|date=July 2024}} Further, a strong Riemannian manifold for which all closed and bounded subsets are compact might not be geodesically complete.{{citation needed|date=July 2024}}

If $g$ is a weak Riemannian metric, then no notion of completeness implies the other in general.{{citation needed|date=July 2024}}

{{div col|colwidth=25em}}

• Smooth manifold
• Riemannian geometry
• Finsler manifold
• Sub-Riemannian manifold
• Pseudo-Riemannian manifold
• Metric tensor
• Hermitian manifold
• Symplectic manifold
• Kahler manifold
• Einstein manifold

{{div col end}}

{{reflist}}

{{refbegin}}

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• {{cite book|mr=2394158|last1=Cheeger|first1=Jeff|zbl=1142.53003|last2=Ebin|first2=David G.|title=Comparison theorems in Riemannian geometry|edition=Revised reprint of the 1975 original|publisher=AMS Chelsea Publishing|location=Providence, RI|year=2008|isbn=978-0-8218-4417-5|author-link1=Jeff Cheeger|author-link2=David Gregory Ebin|doi=10.1090/chel/365}}
• {{Cite book | last1=do Carmo | first1=Manfredo Perdigão |authorlink=Manfredo do Carmo | title=Riemannian geometry | publisher=Birkhäuser Boston | location=Boston, MA | series=Mathematics: Theory & Applications | isbn=978-0-8176-3490-2 |mr=1138207 | year=1992|edition=Translated from the second Portuguese edition of 1979 original|zbl=0752.53001}}
• {{cite book|last1=Gromov|first1=Misha|title-link=Metric Structures for Riemannian and Non-Riemannian Spaces|title=Metric structures for Riemannian and non-Riemannian spaces|edition=Based on the 1981 French original|series=Progress in Mathematics|volume=152|publisher=Birkhäuser Boston, Inc.|location=Boston, MA|year=1999|isbn=0-8176-3898-9|mr=1699320|translator-last1=Bates|translator-first1=Sean Michael|others=With appendices by M. Katz, P. Pansu, and S. Semmes.|doi=10.1007/978-0-8176-4583-0|zbl=0953.53002|author-link1=Mikhael Gromov (mathematician)}}
• {{cite book|author-link1=Shoshichi Kobayashi|author-link2=Katsumi Nomizu|last1=Kobayashi|first1=Shoshichi|last2=Nomizu|title-link=Foundations of Differential Geometry|first2=Katsumi|title=Foundations of differential geometry. Volume I |publisher=John Wiley & Sons, Inc.|location=New York–London|year=1963|mr=0152974|zbl=0119.37502}}
• {{cite book|author-last=Lee|author-first=John M.|author-link=John M. Lee|title=Introduction to Riemannian Manifolds|date=2018|publisher=Springer-Verlag|isbn=978-3-319-91754-2}}
• {{cite book|last1=Petersen|first1=Peter|title=Riemannian geometry|edition=Third edition of 1998 original|series=Graduate Texts in Mathematics|volume=171|publisher=Springer, Cham|year=2016|isbn=978-3-319-26652-7|mr=3469435|doi=10.1007/978-3-319-26654-1|zbl=1417.53001}}
• {{cite book | last = Wolf | first = Joseph A. | edition = Sixth edition of 1967 original | isbn = 978-0-8218-5282-8 | mr = 2742530 | publisher = AMS Chelsea Publishing|location=Providence, RI | title = Spaces of constant curvature | year = 2011|author-link1=Joseph A. Wolf|zbl=1216.53003|doi=10.1090/chel/372}}
• {{cite journal |last1=Burtscher |first1=Annegret |title=Length structures on manifolds with continuous Riemannian metrics |journal=New York Journal of Mathematics |year=2015 |volume=21 |pages=273–296 |issn=1076-9803}}
• {{cite journal |last1=Magnani |first1=Valentino |last2=Tiberio |first2=Daniele |title=A remark on vanishing geodesic distances in infinite dimensions |journal=Proc. Amer. Math. Soc. |date=2020 |volume=148 |issue=1 |pages=3653–3656 |doi=10.1090/proc/14986 |s2cid=204578276 |arxiv=1910.06430 }}
• {{cite journal |arxiv=math/0409303|last1=Michor |first1=Peter W. |last2=Mumford |first2=David |title=Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms |journal=Documenta Mathematica |year=2005 |volume=10 |pages=217–245 |doi=10.4171/dm/187 |doi-access=free |s2cid=69260 }}
• {{cite journal|last1=Milnor|first1=John|author-link1=John Milnor|title=Curvatures of left invariant metrics on Lie groups|journal=Advances in Mathematics|volume=21|pages=293–329|year=1976|issue=3|doi=10.1016/S0001-8708(76)80002-3|doi-access=free|zbl=0341.53030|mr=0425012}}

{{refend}}

• {{springer|id=R/r082180|title=Riemannian metric|author=L.A. Sidorov}}

{{Manifolds}}

{{Riemannian geometry}}

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Category:Riemannian geometry

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Category:Differential geometry